Hide object in a specific area
This is possible, but requires the sphere to be in a separate collection. Assuming it is, you can use the following steps to hide it in one area, while still displaying the object in the other.


Open the sidebar by pressing N and select the View tab in it.
Expand the panel named Collections
Enable Local Collections, this allows you to have different (local) visibility settings per 3D view area.
Hide the collection with the sphere by clicking on the eye icon behind its collection in the panel